Born–Landé equation

The Born–Landé equation is a means of calculating the lattice energy of a crystalline ionic compound. In 1918Max Born and Alfred Landé proposed that the lattice energy could be derived from the electrostatic potential of the ionic lattice and a repulsive potential energy term.

where:

The ionic lattice is modeled as an assembly of hard elastic spheres which are compressed together by the mutual attraction of the electrostatic charges on the ions. They achieve the observed equilibrium distance apart due to a balancing short range repulsion.

The electrostatic potential energy, ${\displaystyle E_{\text{pair}}}$, between a pair of ions of equal and opposite charge is:

where

For a simple lattice consisting ions with equal and opposite charge in a 1:1 ratio, interactions between one ion and all other lattice ions need to be summed to calculate ${\displaystyle E_{M}}$, sometimes called the Madelung or lattice energy:

where

Born and Lande suggested that a repulsive interaction between the lattice ions would be proportional to ${\displaystyle 1/r^{n}}$ so that the repulsive energy term, ${\displaystyle E_{R}}$, would be expressed:

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